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regarding the sets AnsubscriptAnA_{n} from the traveling hump sequence

In this entry, ⌊⋅⌋normal-⋅\lfloor\cdot\rfloor denotes the floor function.

Following is a proof that, for every positiveintegernnn, [n-2⌊log2⁡n⌋2⌊log2⁡n⌋,n-2⌊log2⁡n⌋+12⌊log2⁡n⌋]⊆[0,1]nsuperscript2subscript2nsuperscript2subscript2nnsuperscript2subscript2n1superscript2subscript2n01\displaystyle\left[\frac{n-2^{{\left\lfloor\log_{2}n\right\rfloor}}}{2^{{\left%
\lfloor\log_{2}n\right\rfloor}}},\frac{n-2^{{\left\lfloor\log_{2}n\right%
\rfloor}}+1}{2^{{\left\lfloor\log_{2}n\right\rfloor}}}\right]\subseteq\left[0,%
1\right].

Proof.

Note that this is equivalent to showing that, for every positive integer nnn,

The first inequality is easy to prove: For every positive integer nnn, 2⌊log2⁡n⌋≤2log2⁡n=nsuperscript2subscript2nsuperscript2subscript2nn\displaystyle 2^{{\left\lfloor\log_{2}n\right\rfloor}}\leq 2^{{\log_{2}n}}=n.

Now for the second inequality. Let nnn be a positive integer. Let kkk be the unique positive integer such that